arXiv subjects
Publications and source records attributed to Jonathan Hickman.
This note records an asymptotic improvement on the known $L^p$ range for the Fourier restriction conjecture in high dimensions. This is obtained by combining Guth's polynomial partitioning method with recent geometric results regarding intersections of tubes with nested families of varieties.
The sharp range of $L^p$-estimates for the class of Hörmander-type oscillatory integral operators is established in all dimensions under a general signature assumption on the phase. This simultaneously generalises earlier work of the authors and Guth, which treats the maximal signature case, and also work of Stein and Bourgain--Guth, which treats the minimal signature case.
The sharp range of $L^p$-estimates for the class of Hörmander-type oscillatory integral operators is established in all dimensions under a positive-definite assumption on the phase. This is achieved by generalising a recent approach of the first author for studying the Fourier extension operator, which utilises polynomial partitioning arguments.
We consider Guth's approach to the Fourier restriction problem via polynomial partitioning. By writing out his induction argument as a recursive algorithm and introducing new geometric information, known as the polynomial Wolff axioms, we obtain improved bounds for the restriction conjecture, particularly in high dimensions. Consequences for the Kakeya conjecture are also considered.
We consider a higher dimensional version of the Benjamin--Ono equation, $\partial_t u -\mathcal{R}_1Δu+u\partial_{x_1} u=0$, where $\mathcal{R}_1$ denotes the Riesz transform with respect to the first coordinate. We first establish sharp space--time estimates for the associated linear equation. These estimates enable us to show that the initial value problem for the nonlinear equation is locally well-posed in $L^2$-Sobolev spaces $H^{s}(\mathbb{R}^d)$, with $s>5/3$ if $d=2$ and $s>d/2+1/2$ if $d\ge 3$. We also provide ill-posedness results.
The theory of Fourier integral operators is surveyed, with an emphasis on local smoothing estimates and their applications. After reviewing the classical background, we describe some recent work of the authors which established sharp local smoothing estimates for a natural class of Fourier integral operators. We also show how local smoothing estimates imply oscillatory integral estimates and obtain a maximal variant of an oscillatory integral estimate of Stein. Together with an oscillatory integral counterexample of Bourgain, this shows that our local smoothing estimates are sharp in odd spatial dimensions. Motivated by related counterexamples, we formulate local smoothing conjectures which take into account natural geometric assumptions arising from the structure of the Fourier integrals.
We adapt Guth's polynomial partitioning argument for the Fourier restriction problem to the context of the Kakeya problem. By writing out the induction argument as a recursive algorithm, additional multiscale geometric information is made available. To take advantage of this, we prove that direction-separated tubes satisfy a multiscale version of the polynomial Wolff axioms. Altogether, this yields improved bounds for the Kakeya maximal conjecture in $\mathbb{R}^n$ with $n=5$ or $n\ge 7$ and improved bounds for the Kakeya set conjecture for an infinite sequence of dimensions.
A new range of uniform $L^p$ resolvent estimates is obtained in the setting of the flat torus, improving previous results of Bourgain, Shao, Sogge and Yao. The arguments rely on the $\ell^2$-decoupling theorem and multidimensional Weyl sum estimates.
The sharp Wolff-type decoupling estimates of Bourgain--Demeter are extended to the variable coefficient setting. These results are applied to obtain new sharp local smoothing estimates for wave equations on compact Riemannian manifolds, away from the endpoint regularity exponent. More generally, local smoothing estimates are established for a natural class of Fourier integral operators; at this level of generality the results are sharp in odd dimensions, both in terms of the regularity exponent and the Lebesgue exponent.
We obtain new bounds for the Kakeya maximal conjecture in most dimensions $n<100$, as well as improved bounds for the Kakeya set conjecture when $n=7$ or $9$. For this we consider Guth and Zahl's strengthened formulation of the maximal conjecture, concerning families of tubes that satisfy the polynomial Wolff axioms. Our results give improved estimates for this strengthened formulation when $n =5$ or $n \geq 7$.
The Fourier restriction phenomenon and the size of Kakeya sets are explored in the setting of the ring of integers modulo $N$ for general $N$ and a striking similarity with the corresponding euclidean problems is observed. One should contrast this with known results in the finite field setting.
An $L^2$ Fourier restriction argument of Bak and Seeger is abstracted to the setting of locally compact abelian groups. This is used to prove new restriction estimates for varieties lying in modules over local fields or rings of integers $\mathbb{Z}/N\mathbb{Z}$.
A sharp bound is obtained for the number of ways to express the monomial $X^n$ as a product of linear factors over $\mathbb{Z}/p^α\mathbb{Z}$. The proof relies on an induction-on-scale procedure which is used to estimate the number of solutions to a certain system of polynomial congruences. The method also applies to more general systems of polynomial congruences that satisfy a non-degeneracy hypothesis.
We prove that the maximal operator associated with variable homogeneous planar curves $(t, u t^α)_{t\in \mathbb{R}}$, $α\not=1$ positive, is bounded on $L^p(\mathbb{R}^2)$ for each $p>1$, under the assumption that $u:\mathbb{R}^2 \to \mathbb{R}$ is a Lipschitz function. Furthermore, we prove that the Hilbert transform associated with $(t, ut^α)_{t\in \mathbb{R}}$, $α\not=1$ positive, is bounded on $L^p(\mathbb{R}^2)$ for each $p>1$, under the assumption that $u:\mathbb{R}^2\to \mathbb{R}$ is a measurable function and is constant in the second variable. Our proofs rely on stationary phase methods, $TT^*$ arguments, local smoothing estimates and a pointwise estimate for taking averages along curves.
A set in the Euclidean plane is constructed whose image under the classical Radon transform is Lipschitz in every direction. It is also shown that, under mild hypotheses, for any such set the function which maps a direction to the corresponding Lipschitz constant cannot be bounded.
Numerous authors have considered the problem of determining the Lebesgue space mapping properties of the operator $\mathcal{A}$ given by convolution with affine arc-length measure on some polynomial curve in Euclidean space. Essentially, $\mathcal{A}$ takes weighted averages over translates of the curve. In this paper a variant of this problem is discussed where averages over both translates and dilates of a fixed curve are considered. The sharp range of estimates for the resulting operator is obtained in all dimensions, except for an endpoint. The techniques used are redolent of those previously applied in the study of $\mathcal{A}$. In particular, the arguments are based upon the refinement method of Christ, although a significant adaptation of this method is required to fully understand the additional smoothing afforded by averaging over dilates.
A Fourier restriction estimate is obtained for a broad class of conic surfaces by adding a weight to the usual underlying measure. The new restriction estimate exhibits a certain affine-invariance and implies the sharp $L^p-L^q$ restriction theorem for compact subsets of a type $k$ conical surface, up to an endpoint. Furthermore, the chosen weight is shown to be, in some quantitative sense, optimal. Appended is a discussion of type k conical restriction theorems which addresses some anomalies present in the existing literature.